mathematics - utah state board of education

UTAH CORE STANDARDS

UTAH STATE OFFICE OF EDUCATION 250 EAST 500 SOUTH P.O. BOX 144200 SALT LAKE CITY, UTAH 84114-4200 SYDNEE DICKSON, INTERIM STATE SUPERINTENDENT OF PUBLIC INSTRUCTION

MATHEMATICSMiddle/Junior High (6–8)

INTRO

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UTAH CORE STATE STANDARDSfor

MATHEMATICSMIDDLE/JUNIOR HIGH SCHOOL

GRADES (6–8)

Adopted August 2010

by the

Utah State Board of Education

Utah EducationSTATEOFFICE

of

Revised September 2015–April 2016

The Utah State Board of Education, in January of 1984, established policy requiring the identification of specific core standards to be met by all K–12 students in order to graduate from Utah’s secondary schools. The Utah State Board of Education regularly updates the Utah Core Standards, while parents, teachers, and local school boards continue to control the curriculum choices that reflect local values.

The Utah Core Standards are aligned to scientifically based content standards. They drive high quality instruction through statewide comprehensive expectations for all students. The standards outline essential knowledge, concepts, and skills to be mastered at each grade level or within a critical content area. The standards provide a foundation for ensuring learning within the classroom.

UTAH CORE STATE STANDARDS for MATHEMATICSPREFACE | v

UTAH STATE BOARDOF EDUCATION

District Name City

District 1 Terryl Warner Hyrum, UT 84319

District 2 Spencer F. Stokes Ogden, UT 84403

District 3 Linda B. Hansen West Valley City, UT 84120

District 4 David L. Thomas South Weber, UT 84405

District 5 Laura Belnap Bountiful, UT 84010

District 6 Brittney Cummins West Valley City, UT 84120

District 7 Leslie B. Castle Salt Lake City, UT 84108

District 8 Jennifer A. Johnson Murray, UT 84107

District 9 Joel Wright Cedar Hills, UT 84062

District 10 David L. Crandall Draper, UT 84020

District 11 Jefferson Moss Saratoga Springs, UT 84045

District 12 Dixie L. Allen Vernal, UT 84078

District 13 Stan Lockhart Provo, UT 84604

District 14 Mark Huntsman Fillmore, UT 84631

District 15 Barbara W. Corry Cedar City, UT 84720

Sydnee Dickson Interim State Superintendent of Public Instruction

Lorraine Austin Secretary to the Board

2/2016

250 East 500 South P. O. Box 144200 Salt Lake City, UT 84114-4200http://schoolboard.utah.gov

UTAH CORE STATE STANDARDS for MATHEMATICS

PREFACE | vi

UTAH CORE STATE STANDARDS for MATHEMATICSPREFACE | vii

Table of Contents

Introduction ix

Mathematics Middle/Junior High Courses (6–8) 1Grade 6 3Grade 7 13Grade 8 21

UTAH CORE STATE STANDARDS for MATHEMATICSPREFACE | ix

Organization of the StandardsThe Utah Core Standards are organized into strands, which represent significant areas of learning within content areas. Depending on the core area, these strands may be designated by time periods, thematic principles, modes of practice, or other organizing principles.

Within each strand are standards. A standard is an articulation of the demonstrated proficien-cy to be obtained. A standard represents an essential element of the learning that is expected. While some standards within a strand may be more comprehensive than others, all standards are essential for mastery.

Understanding MathematicsThese standards define what students should understand and be able to do in their study of mathematics. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in a way appropriate to the student's mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

The standards set grade-specific standards but do not dictate curriculum or teaching meth-ods, nor do they define the intervention methods or materials necessary to support students who are well below or well above grade-level expectations. It is also beyond the scope of the Standards to define the full range of supports appropriate for English language learners and for students with special needs. At the same time, all students must have the opportunity to learn and meet the same high standards if they are to access the knowledge and skills neces-sary in their post-school lives. The standards should be read as allowing for the widest pos-sible range of students to participate fully from the outset, along with appropriate accommo-dations to ensure maximum participation of students with special education needs. No set of grade-specific standards can fully reflect the great variety in abilities, needs, learning rates, and achievement levels of students in any given classroom. However, the standards do pro-vide clear signposts along the way to the goal of college and career readiness for all students.

What students can learn at any particular grade level depends upon what they have learned before. Ideally then, each standard in this document might have been phrased in the form, "Students who already know… should next come to learn ..." Grade placements for specific topics have been made on the basis of state and international comparisons and the collective experience and collective professional judgment of educators, researchers and mathemati-cians. Learning opportunities will continue to vary across schools and school systems, and educators should make every effort to meet the needs of individual students based on their current understanding.

INTRODUCTION

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UTAH CORE STATE STANDARDSfor

MATHEMATICS

6–8Utah EducationSTATE

OFFICEof

UTAH CORE STATE STANDARDS for MATHEMATICSG

RADE 6 | 3

Mathematics | Grade 6In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.

(1) Students will use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students will connect their under-standing of multiplication and division with ratios and rates. Thus students will expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students will solve a wide variety of problems in-volving ratios and rates.

(2) Students will use the meaning of fractions, the meanings of multiplication and divi-sion, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students will use these operations to solve problems. Students will extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which includes negative ratio-nal numbers, and in particular negative integers. They will reason about the order and ab-solute value of rational numbers and about the location of points in all four quadrants of the coordinate plane.

(3) Students will understand the use of variables in mathematical expressions. They will write expressions and equations that correspond to given situations, evaluate expressions, and use expressions and formulas to solve problems. Students will understand that ex-pressions in different forms can be equivalent, and they will use the properties of opera-tions to rewrite expressions in equivalent forms. Students will know that the solutions of an equation are the values of the variables that make the equation true. Students will use properties of operations and the idea of maintaining the equality of both sides of an equa-tion to solve simple one-step equations. Students will construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they will use equations (such as 3x = y) to describe relationships between quantities.

(4) Building on and reinforcing their understanding of number, students will begin to de-velop their ability to think statistically. Students will recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point. Students will recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different


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sets of data can have the same mean and median yet be distinguished by their variability. Students will learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected.

(5) Students in Grade 6 will also build on their work with area in elementary school by rea-soning about relationships among shapes to determine area, surface area, and volume. They will find areas of right triangles, other triangles, and special quadrilaterals by de-composing these shapes, rearranging or removing pieces, and relating the shapes to rect-angles. Using these methods, students will discuss, develop, and justify formulas for areas of triangles and parallelograms. Students will find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They will reason about right rectangular prisms with fractional side lengths to extend for-mulas for the volume of a right rectangular prism to fractional side lengths. They will pre-pare for work on scale drawings and constructions in Grade 7 by drawing polygons in the coordinate plane.


RADE 6 | 5

Strand: MATHEMATICAL PRACTICES (6.MP)

The Standards for Mathematical Practice in Sixth Grade describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically pro-ficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes (Standards 6.MP.1–8).

� Standard 6.MP.1 Make sense of problems and persevere in solving them. Explain the meaning of a problem and look for entry points to its solution. Analyze givens, con-straints, relationships, and goals. Make conjectures about the form and meaning of the solution, plan a solution pathway, and continually monitor progress asking, “Does this make sense?” Consider analogous problems, make connections between multiple repre-sentations, identify the correspondence between different approaches, look for trends, and transform algebraic expressions to highlight meaningful mathematics. Check an-swers to problems using a different method.

� Standard 6.MP.2 Reason abstractly and quantitatively. Make sense of the quantities and their relationships in problem situations. Translate between context and algebraic representations by contextualizing and decontextualizing quantitative relationships. This includes the ability to decontextualize a given situation, representing it algebra-ically and manipulating symbols fluently as well as the ability to contextualize algebraic representations to make sense of the problem.

� Standard 6.MP.3 Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of state-ments to explore the truth of their conjectures. Justify conclusions and communicate them to others. Respond to the arguments of others by listening, asking clarifying ques-tions, and critiquing the reasoning of others.

� Standard 6.MP.4 Model with mathematics. Apply mathematics to solve problems aris-ing in everyday life, society, and the workplace. Make assumptions and approximations, identifying important quantities to construct a mathematical model. Routinely interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

� Standard 6.MP.5 Use appropriate tools strategically. Consider the available tools and be sufficiently familiar with them to make sound decisions about when each tool might be helpful, recognizing both the insight to be gained as well as the limitations. Identify relevant external mathematical resources and use them to pose or solve problems. Use tools to explore and deepen their understanding of concepts.

� Standard 6.MP.6 Attend to precision. Communicate precisely to others. Use explicit definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the corre-spondence with quantities in a problem. Calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context.


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� Standard 6.MP.7 Look for and make use of structure. Look closely at mathematical re-lationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expres-sions, as single objects or as being composed of several objects. For example, see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

� Standard 6.MP.8 Look for and express regularity in repeated reasoning. Notice if rea-soning is repeated, and look for both generalizations and shortcuts. Evaluate the reason-ableness of intermediate results by maintaining oversight of the process while attending to the details.

Strand: RATIOS AND PROPORTIONAL RELATIONSHIPS (6.RP)

Understand ratio concepts and use ratio reasoning to solve problems (Standards 6.RP.1–3).

� Standard 6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. The following are examples of ratio language: “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every two wings there was one beak.” “For every vote candidate A received, candidate C received nearly three votes.”

� Standard 6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. The following are examples of rate language: "This recipe has a ratio of four cups of flour to two cups of sugar, so the rate is two cups of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (In sixth grade, unit rates are limited to non-complex fractions.)

� Standard 6.RP.3 Use ratio and rate reasoning to solve real-world (with a context) and mathematical (void of context) problems, using strategies such as reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations involving unit rate problems.

a. Make tables of equivalent ratios relating quantities with whole-number measure-ments, find missing values in the tables, and plot the pairs of values on the coordi-nate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took four hours to mow eight lawns, how many lawns could be mowed in 32 hours? What is the hourly rate at which lawns were being mowed?

c. Find a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part and the percent. (For example, 30% of a quantity means 30/100 times the quantity.)

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.


RADE 6 | 7

Strand: THE NUMBER SYSTEM (6.NS)

Apply and extend previous understandings of multiplication and division of whole numbers to divide fractions by fractions (Standard 6.NS.1). Compute (add, subtract, multiply and di-vide) fluently with multi-digit numbers and decimals and find common factors and multiples (Standards 6.NS.2–4). Apply and extend previous understandings of numbers to the system of rational numbers (Standards 6.NS.5–8).

� Standard 6.NS.1 Interpret and compute quotients of fractions.

a. Compute quotients of fractions by fractions, for example, by applying strategies such as visual fraction models, equations, and the relationship between multiplication and division, to represent problems.

b. Solve real-world problems involving division of fractions by fractions. For example, how much chocolate will each person get if three people share 1/2 pound of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectan-gular strip of land with length 3/4 mile and area 1/2 square mile?

c. Explain the meaning of quotients in fraction division problems. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.)

� Standard 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

a. Fluently divide multi-digit decimals using the standard algorithm, limited to a whole number dividend with a decimal divisor or a decimal dividend with a whole number divisor.

b. Solve division problems in which both the dividend and the divisor are multi-digit decimals; develop the standard algorithm by using models, the meaning of division, and place value understanding.

� Standard 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

� Standard 6.NS.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9 + 2).

� Standard 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (for example, temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world con-texts, explaining the meaning of zero in each situation.


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� Standard 6.NS.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of zero on the number line; recognize that the opposite of the opposite of a number is the number itself. For example, -(-3) = 3, and zero is its own opposite.

b. Understand that the signs of numbers in ordered pairs indicate their location in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

� Standard 6.NS.7 Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3° C > –7° C to express the fact that –3° C is warmer than –7° C.

c. Understand the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as magnitude for a positive or negative quan-tity in a real-world context. For example, for an account balance of –30 dollars, write |–30|= 30 to describe the size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. For example: Recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

� Standard 6.NS.8 Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same x-coordinate or the same y-coordinate.

Strand: EXPRESSIONS AND EQUATIONS (6.EE)

Apply and extend previous understandings of arithmetic to algebraic expressions involving exponents and variables (Standards 6.EE.1–4). They reason about and solve one-variable equations and inequalities (Standards 6.EE.5–8). Represent and analyze quantitative rela-tionships between dependent and independent variables in a real-world context (Standard 6.EE.9).

� Standard 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.


RADE 6 | 9

� Standard 6.EE.2 Write, read, and evaluate expressions in which letters represent numbers.

a. Write expressions that record operations with numbers and with letters representing numbers. For example, express the calculation "Subtract y from 5" as 5 – y and express “Jane had $105.00 in her bank account. One year later, she had x dollars more. Write an expression that shows her new balance” as $105.00 + x.

b. Identify parts of an expression using mathematical terms (for example, sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity and a sum of two terms. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, in-cluding those involving whole-number exponents, applying the Order of Operations when there are no parentheses to specify a particular order. For example, use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2.

� Standard 6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equiva-lent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

� Standard 6.EE.4 Identify when two expressions are equivalent. For example, the expressions y + y + y and 3y are equivalent because they name the same number, regardless of which number y represents.

� Standard 6.EE.5 Understand solving an equation or inequality as a process of answering the question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

� Standard 6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an un-known number, or, depending on the purpose at hand, any number in a specified set.

� Standard 6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + a = b and ax = b for cases in which a, b and x are all non-nega-tive rational numbers.

� Standard 6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

� Standard 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity,


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thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and indepen-dent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Strand: GEOMETRY (6.G)

Solve real-world and mathematical problems involving area, surface area, and volume (Standards 6.G.1–4).

� Standard 6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing and decomposing into rectangles, triangles and/or other shapes; apply these techniques in the context of solving real-world and mathematical problems.

� Standard 6.G.2 Find the volume of a right rectangular prism with appropriate unit frac-tion edge lengths by packing it with cubes of the appropriate unit fraction edge lengths (for example, 3½ x 2 x 6), and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solv-ing real-world and mathematical problems. (Note: Model the packing using drawings and diagrams.)

� Standard 6.G.3 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same x coordinate or the same y coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

� Standard 6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

Strand: STATISTICS AND PROBABILITY (6.SP)

Develop understanding of statistical variability of data (Standards 6.SP.1–3). Summarize and describe distributions (Standards 6.SP.4–5).

� Standard 6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students’ ages.

� Standard 6.SP.2 Understand that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread/range and overall shape.


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� Standard 6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its val-ues vary with a single number.

� Standard 6.SP.4 Display numerical data in plots on a number line, including dot plots, his-tograms and box plots. Choose the most appropriate graph/plot for the data collected.

� Standard 6.SP.5 Summarize numerical data sets in relation to their context, such as by:

a. Reporting the number of observations.

b. Describing the nature of the attribute under investigation, including how it was mea-sured and its units of measurement.

c. Giving quantitative measures of center (median and/or mean) and variability (inter-quartile range and/or mean absolute deviation), as well as describing any overall pat-tern and any striking deviations (for example, outliers) from the overall pattern with reference to the context in which the data were gathered.

d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.


RADE 7 | 13

Mathematics | Grade 7In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

(1) Students extend their understanding of ratios and develop understanding of propor-tionality to solve single- and multi-step problems. Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar ob-jects. Students graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish propor-tional relationships from other relationships.

(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different repre-sentations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational numbers, maintaining the properties of operations and the relation-ships between addition and subtraction, and between multiplication and division. By ap-plying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one vari-able and use these equations to solve problems.

(3) Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of three-dimensional objects. In prepa-ration for work on congruence and similarity in Grade 8, they reason about relationships among two-dimensional figures using scale drawings and informal geometric construc-tions, and they gain familiarity with the relationships between angles formed by intersect-ing lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining cross-sections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.

(4) Students build on their previous work with single-data distributions to compare two-data distributions and address questions about differences between populations. They begin informal work with random sampling to generate data sets and learn about the im-portance of representative samples for drawing inferences.


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The Standards for Mathematical Practice in Seventh Grade describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematical-ly proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes (Standards 7.MP.1–8).






� Standard 7.MP.6 Attend to precision. Communicate precisely to others. Use explicit definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the cor-respondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.


RADE 7 | 15

� Standard 7.MP.7 Look for and make use of structure. Look closely at mathematical relationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.


Strand: RATIOS AND PROPORTIONAL RELATIONSHIPS (7.RP)

Analyze proportional relationships and use them to solve real-world and mathematical prob-lems (Standards 7.RP.1–3).

� Standard 7.RP.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.

� Standard 7.RP.2 Recognize and represent proportional relationships between quantities.

a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing wheth-er the graph is a straight line through the origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, dia-grams, and verbal descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is pro-portional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

� Standard 7.RP.3 Use proportional relationships to solve multi-step ratio and percent prob-lems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Strand: THE NUMBER SYSTEM (7.NS)

Apply and extend previous understandings of operations with fractions to add, subtract, mul-tiply, and divide rational numbers (Standards 7.NS.1–3).


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� Standard 7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p in the positive or nega-tive direction, depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

� Standard 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

a. Understand that multiplication is extended from fractions to rational numbers by re-quiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided the divisor is not zero, and that every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.

d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

� Standard 7.NS.3 Solve real-world and mathematical problems involving the four opera-tions with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions.


Use properties of operations to generate equivalent expressions (Standards 7.EE.1–2). Solve real-life and mathematical problems using numerical and algebraic expressions and equa-tions (Standards 7.EE.3–4).

� Standard 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.


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� Standard 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem, and how the quantities in it are related. For ex-ample, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

� Standard 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate, and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

� Standard 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reason-ing about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluent-ly. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.


Draw, construct, and describe geometrical figures, and describe the relationships between them (Standards 7.G.1–3). Solve real-life and mathematical problems involving angle mea-sure, area, surface area, and volume (Standards 7.G.4–6).

� Standard 7.G.1 Solve problems involving scale drawings of geometric figures, such as computing actual lengths and areas from a scale drawing and reproducing a scale draw-ing at a different scale.

� Standard 7.G.2 Draw (freehand, with ruler and protractor, and with technology) geomet-ric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

� Standard 7.G.3 Describe the two-dimensional figures that result from slicing three-di-mensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.


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� Standard 7.G.4 Know the formulas for the area and circumference of a circle, and solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

� Standard 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write, and use them to solve simple equations for an unknown angle in a figure.

� Standard 7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilater-als, polygons, cubes, and right prisms.


Use random sampling to draw inferences about a population (Standards 7.SP.1–2). Draw informal comparative inferences about two populations (Standards 7.SP.3–4). Investigate chance processes and develop, use, and evaluate probability models (Standards 7.SP.5–8).

� Standard 7.SP.1 Understand that statistics can be used to gain information about a popu-lation by examining a sample of the population, and that generalizations about a popu-lation from a sample are valid only if the sample is representative of that population. Understand that random sampling is more likely to produce representative samples and support valid inferences.

� Standard 7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; pre-dict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

� Standard 7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, estimating the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soc-cer team, approximately twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

� Standard 7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh grade science book are gen-erally longer than the words in a chapter of a fourth grade science book.

� Standard 7.SP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.


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� Standard 7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and pre-dict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

� Standard 7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probabili-ty that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

� Standard 7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

a. Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event.

c. Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the ques-tion: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?


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Mathematics | Grade 8In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

(1) Students use linear equations, linear inequalities, and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for pro-portions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the in-put or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m × A. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation.

Students strategically choose and efficiently implement procedures to solve lin-ear equations in one variable, understanding that when they use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, linear inequalities, systems of lin-ear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.

(2) Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity de-termines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial repre-sentations), and they describe how aspects of the function are reflected in the different representations.

(3) Students use ideas about distance and angles; how they behave under translations, rotations, reflections, and dilations; and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configura-tions of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decom-posing a square in two different ways. They apply the Pythagorean Theorem to find dis-tances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.


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The Standards for Mathematical Practice in Eighth Grade describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematical-ly proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes (Standards 8.MP.1–8).






� Standard 8.MP.6 Attend to precision. Communicate precisely to others. Use explicit definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the cor-respondence with quantities in a problem. Calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.


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� Standard 8.MP.7 Look for and make use of structure. Look closely at mathematical re-lationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expres-sions, as single objects or as being composed of several objects. For example, see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.


Strand: NUMBER SYSTEM (8.NS)

Know that there are numbers that are not rational, and approximate them by rational num-bers (Standards 8.NS.1–3).

� Standard 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers, show that the decimal expansion repeats eventually, and convert a decimal expansion which re-peats eventually into a rational number.

� Standard 8.NS.2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., ∏ 2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

� Standard 8.NS.3 Understand how to perform operations and simplify radicals with empha-sis on square roots.


Work with radical and integer exponents (Standards 8.EE.1–4). Understand the connec-tions between proportional relationships, lines, and linear relationships (Standards 8.EE.5–6). Analyze and solve linear equations and inequalities and pairs of simultaneous linear equations (Standards 8.EE.7–8).

� Standard 8.EE.1 Know and apply the properties of integer exponents to generate equiva-lent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

� Standard 8.EE.2 Use square root and cube root symbols to represent solutions to equa-tions of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

� Standard 8.EE.3 Use numbers expressed in the form of a single digit times an integer


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power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

� Standard 8.EE.4 Perform operations with numbers expressed in scientific notation, includ-ing problems where both decimal and scientific notation are used. Use scientific nota-tion and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific nota-tion that has been generated by technology.

� Standard 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

� Standard 8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

� Standard 8.EE.7 Solve linear equations and inequalities in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successive-ly transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

b. Solve single-variable linear equations and inequalities with rational number coef-ficients, including equations and inequalities whose solutions require expanding expressions using the distributive property and collecting like terms.

c. Solve single-variable absolute value equations.

� Standard 8.EE.8 Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equations in two variables corre-spond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables graphically, approximating when solutions are not integers and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables graphically. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.


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Strand: FUNCTIONS (8.F)

Define, evaluate, and compare functions (Standards 8.F.1–3). Use functions to model rela-tionships between quantities (Standards 8.F.4–5).

� Standard 8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in grade 8.)

� Standard 8.F.2 Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

� Standard 8.F.3 Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2, giving the area of a square as a function of its side length, is not linear be-cause its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

� Standard 8.F.4 Construct a function to model a linear relationship between two quanti-ties. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

� Standard 8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or non-linear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.


Understand congruence and similarity using physical models, transparencies, or geometry software (Standards 8.G.1–5). Understand and apply the Pythagorean Theorem and its con-verse (Standards 8.G.6–8). Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres (Standard 8.G.9).

� Standard 8.G.1 Verify experimentally the properties of rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

� Standard 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and trans-lations; given two congruent figures, describe a sequence that exhibits the congruence between them.


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� Standard 8.G.3 Observe that orientation of the plane is preserved in rotations and trans-lations, but not with reflections. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

� Standard 8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, transla-tions, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

� Standard 8.G.5 Use informal arguments to establish facts about the angle sum and ex-terior angle of triangles, about the angles created when parallel lines are cut by a trans-versal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

� Standard 8.G.6 Explore and explain proofs of the Pythagorean Theorem and its converse.

� Standard 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

� Standard 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

� Standard 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.


Investigate patterns of association in bivariate data (Standards 8.SP.1–4).

� Standard 8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

� Standard 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

� Standard 8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. (Calculating equations for a linear model is not expected in grade 8.)

� Standard 8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table.


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Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or col-umns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

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